We develop first-order smoothing techniques for saddle-point problems that arise in finding a Nash equilibrium of sequential games. The crux of our work is a construction of suitable prox-functions for a certain class of polytopes that encode the sequential nature of the game. We also introduce heuristics that significantly speed up the algorithm, and decomposed game representations that reduce the memory requirements, enabling the application of the techniques to drastically larger games. An implementation based on our smoothing techniques computes approximate Nash equilibria for games that are more than four orders of magnitude larger than what prior approaches can handle. Finally, we show near-linear further speedups from parallelization.

We address the feasibility (existence of non-trivial solutions) of the pair of alternative conic systems of constraintswhere A ∈ R m×n , m < n, is a full row-rank matrix, and C ⊆ R n is a closed convex cone. To this end, we reformulate the above pair of conic systems as a primal-dual pair of conic programs. Each of the conic programs corresponds to a natural relaxation of each of the two conic systems.When C is a self-scaled cone with a known self-scaled barrier, the conic programming reformulation can be solved via an interior-point algorithm. For a well-posed instance A, a strict solution to one of the two original conic systems can be obtained in O( √ C log( C C(A)) interior-point iterations. Here C is the complexity parameter of the self-scaled barrier of C and C(A) is Renegar's condition number of A. A central feature of our approach is the conditioning of the system of equations that arise at each interior-point iteration. The condition number of such system of equations grows in a controlled manner and remains bounded by a constant factor of C(A) 2 throughout the entire algorithm.

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